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Qubit-excitation-based adaptive variational quantum eigensolver

Yordan S. Yordanov, V. Armaos, Crispin H. W. Barnes, David R. M. Arvidsson-Shukur

arXiv:2011.10540v2quant-ph

TL;DR

Molecular VQE needs ansätze that accurately approximate ground states, are easy to optimize, and use shallow circuits; this paper introduces QEB-ADAPT-VQE, which grows an ansatz using qubit-excitation evolutions. At higher accuracies, it outperforms qubit-ADAPT-VQE in CNOT efficiency, though qubit-ADAPT-VQE is more efficient at low accuracies.

  • Problem

    Molecular VQE requires ansätze that accurately approximate the ground state, are easy to optimize, and can be implemented with shallow circuits.

    Source: §II · p. 2

  • Method

    QEB-ADAPT-VQE iteratively grows a problem-tailored molecular ansatz by appending unitary evolutions of qubit-excitation operators.

    Source: §II · p. 3

  • Results

    At higher accuracies, QEB-ADAPT-VQE systematically outperformed qubit-ADAPT-VQE in CNOT efficiency, while qubit-ADAPT-VQE was more efficient at low accuracies.

    Source: §IV · p. 16

  • Takeaways & Limitations

    Qubit-excitation evolutions can construct accurate molecular ansätze with hardware efficiency despite lacking some physical features of fermionic excitation evolutions.

    Source: §II · p. 3

    Using more rudimentary unitary operations can require additional variational parameters and iterations to reach a given accuracy.

    Source: §II · p. 3

Abstract

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Molecular simulations with the variational quantum eigensolver (VQE) are a promising application for emerging noisy intermediate-scale quantum computers. Constructing accurate molecular ansätze that are easy to optimize and implemented by shallow quantum circuits is crucial for the successful implementation of such simulations. Ansätze are, generally, constructed as series of fermionic-excitation evolutions. Instead, we demonstrate the usefulness of constructing ansätze with "qubit-excitation evolutions", which, contrary to fermionic excitation evolutions, obey "qubit commutation relations". We show that qubit excitation evolutions, despite the lack of some of the physical features of fermionic excitation evolutions, accurately construct ansätze, while requiring asymptotically fewer gates. Utilizing qubit excitation evolutions, we introduce the qubit-excitation-based adaptive (QEB-ADAPT)-VQE protocol. The QEB-ADAPT-VQE is a modification of the ADAPT-VQE that performs molecular simulations using a problem-tailored ansatz, grown iteratively by appending evolutions of qubit excitation operators. By performing classical numerical simulations for small molecules, we benchmark the QEB-ADAPT-VQE, and compare it against the original fermionic-ADAPT-VQE and the qubit-ADAPT-VQE. In terms of circuit efficiency and convergence speed, we demonstrate that the QEB-ADAPT-VQE outperforms the qubit-ADAPT-VQE, which to our knowledge was the previous most circuit-efficient scalable VQE protocol for molecular simulations.

II. INTRODUCTION

The paper introduces QEB-ADAPT-VQE, which builds molecular ansätze from qubit excitation evolutions to combine high accuracy with circuit efficiency. It aims to avoid the extra parameters and iterations required by qubit-ADAPT-VQE while using circuits acting on fixed numbers of qubits.

VQE background

  • VQE estimates a Hamiltonian’s lowest eigenvalue with a parameterized quantum state and uses shallower circuits than quantum phase estimation, improving noise resistance at the cost of more measurements and classical post-processing [6–14].The introduction presents VQE as a promising approach for molecular electronic-structure simulations on noisy intermediate-scale quantum computers [5].Source: §II · pp. 1–2

Molecular ansätze

  • UCC ansätze respect electronic-wavefunction symmetries and are accurate for weakly correlated systems, but redundant terms and higher-order excitations or Trotterization add overhead.The paper frames molecular VQE ansatz design around accuracy, ease of optimization, and shallow circuits.Source: §II · p. 2
  • Hardware-efficient ansätze offer shallow circuits and flexibility, but their large parameter counts and potentially vanishing energy gradients can make optimization intractable for large molecules [27] [31] [33] [34] [35].The passage identifies the vanishing-gradient scenario as the barren-plateau problem [33] [34] [35].Source: §II · p. 2

Prior iterative VQE protocols

  • Qubit-ADAPT-VQE creates shallower circuits than fermionic-ADAPT-VQE, but requires additional variational parameters and iterations to reach a given accuracy.Fermionic-ADAPT-VQE achieved chemical accuracy (10−3 Hartree) with several times fewer variational parameters than UCCSD.Source: §II · pp. 2–3

QEB-ADAPT-VQE

  • Qubit excitation evolutions enable QEB-ADAPT-VQE to build problem-tailored molecular ansätze with fixed-qubit circuits and more rapid construction than Pauli string exponentials.The authors report that these evolutions can approximate electronic wavefunctions almost as accurately as fermionic excitation evolutions, and describe a modified ansatz-growing strategy that increases quantum measurements by a constant factor.Source: §II · p. 3

III. RESULTS · A. Theoretical background and notation

The section establishes the electronic-structure eigenvalue problem and notation, then describes the Jordan–Wigner mapping and the VQE workflow used to estimate molecular ground-state energies.

A. Theoretical background and notation

  • The electronic structure problem is to find a molecule’s or atom’s ground-state electron wavefunction and energy by solving the time-independent Schrödinger equation [15].Within the Born–Oppenheimer approximation, the nuclei are treated as motionless and the electronic Hamiltonian is written in second-quantized form.Source: A. Theoretical background and notation · p. 3
  • NMO denotes the number of molecular spin-orbitals; a† and a are fermionic creation and annihilation operators, and h terms are one- and two-electron integrals.Source: A. Theoretical background and notation · p. 3
  • The Hamiltonian can be mapped to quantum-gate operators using Jordan–Wigner or Bravyi–Kitaev encoding; this work assumes Jordan–Wigner, mapping each spin-orbital’s occupancy to its qubit state.Source: A. Theoretical background and notation · pp. 3–4
  • Under Jordan–Wigner, Pauli-Z strings compute state parity and supply exchange phase factors that account for fermionic anticommutation.The passage identifies the fermionic creation and annihilation operators as satisfying anticommutation relations.Source: A. Theoretical background and notation · p. 4
  • The Pauli-operator Hamiltonian contains a number of terms that scales as O(N^4).Its terms use Pauli operators acting on qubits with real scalar coefficients.Source: A. Theoretical background and notation · p. 4
  • The VQE workflow estimates molecular ground-state energy by minimizing the trial-state energy under the Rayleigh–Ritz variational principle.A quantum computer prepares the trial state and evaluates its energy; a classical computer processes measurement data and updates the parameters.Source: A. Theoretical background and notation · p. 4
  • The VQE trial state is generated by applying the parameterized ansatz U(θ) to an initial reference state.Source: A. Theoretical background and notation · p. 4

B. The ADAPT-VQE protocols

ADAPT-VQE iteratively builds a system-tailored ansatz by adding unitary elements and reoptimizing its parameters. Fermionic- and qubit-ADAPT-VQE use different element pools but select additions by energy-gradient magnitude.

B. The ADAPT-VQE protocols

  • At each iteration, ADAPT-VQE appends one or more unitary ansatz elements from a finite pool and optimizes the expanded parameter set with VQE.The ansatz-growing strategy aims to achieve the lowest estimate of E(θ(m)).Source: B. The ADAPT-VQE protocols · pp. 4–5
  • The greedy iterations tailor the ansatz to the simulated system, potentially using fewer variational parameters and a shallower circuit than fixed general-purpose ansätze such as UCCSD.Source: B. The ADAPT-VQE protocols · p. 5
  • Fermionic-ADAPT-VQE uses spin-complement pairs of single and double fermionic excitation evolutions, whereas qubit-ADAPT-VQE uses parametrized exponentials of XY-Pauli strings; both add the element with the largest energy-gradient magnitude.Source: B. The ADAPT-VQE protocols · p. 5

C. Ansatz elements

Qubit-excitation evolutions use circuits with fixed CNOT counts, unlike fermionic-excitation circuits whose costs scale with system size. They also reduce variational parameters relative to separate Pauli-string exponentials and permit local circuit optimizations.

C. Ansatz elements

  • Fermionic-excitation circuit size scales as O(NMO); in Jordan-Wigner encoding, qubit excitations become increasingly CNOT-efficient for larger systems.Fermionic single and double evolutions have CNOT counts 2(k −i) + 1 and 2(l + j −i −k) + 9, respectively.Source: C. Ansatz elements · pp. 6–7
  • Single qubit-excitation evolutions require 2 CNOTs and double qubit-excitation evolutions require 13, with both counts fixed.The corresponding circuits are shown in Figs. 1 and 2.Source: C. Ansatz elements · p. 6, Figure 1 · p. 6, Figure 2 · p. 7
  • Single and double qubit-excitation evolutions combine 2 and 8 mutually commuting Pauli-string exponentials, respectively, reducing variational parameters and enabling local circuit optimizations unavailable to separate Pauli-string exponentials.The local optimizations are those of Ref..Source: C. Ansatz elements · p. 7
  • For qubit-ADAPT-VQE comparisons, the paper assumes standard CNOT staircase circuits for length-l Pauli-string exponentials, with 2(l −1) CNOTs; global circuit optimization is excluded.The QEB-ADAPT-VQE and fermionic-ADAPT-VQE circuits are assumed to use circuits derived in Ref..Source: C. Ansatz elements · p. 7

D. The QEB-ADAPT-VQE protocol

QEB-ADAPT-VQE builds a molecular ansatz by selecting qubit-excitation evolutions, using measured energy gradients to screen candidates and optimized energy reductions to choose additions. For the tested molecules, screening multiple candidates reduced CNOT counts, while optionally appending spin complements roughly halved the iterations needed for a target accuracy.

D. The QEB-ADAPT-VQE protocol

  • The protocol transforms the molecular Hamiltonian into a gate-operator representation, defines a pool of unique single and double qubit-excitation evolutions, and uses the Hartree-Fock state in its simulations.The required one- and two-electron integrals can be calculated classically in time polynomial in NMO.Source: D. The QEB-ADAPT-VQE protocol · pp. 7–8
  • At each iteration, QEB-ADAPT-VQE measures gradients at zero parameter values via Hamiltonian–excitation commutator expectations, then optimizes the top-gradient candidates and selects the largest energy reduction.The zero-parameter evaluation assumes the reference state is close to the ground state; selecting among n > 1 candidates uses more measurements than direct single-candidate selection.Source: D. The QEB-ADAPT-VQE protocol · pp. 8–9
  • When the ground state is known a priori to have the reference state's spin, appending a spin-complement evolution with an independent parameter roughly halves the iterations needed for a given accuracy.Independent parameters provide flexibility to obtain the correct relative sign because qubit-excitation evolutions do not account for state parity.Source: D. The QEB-ADAPT-VQE protocol · p. 9

E. Classical numerical simulations

Classical VQE simulations on LiH, H6, and BeH2 compare qubit and fermionic excitations and benchmark QEB-ADAPT-VQE. The simulations use STO-3G without frozen orbitals and analytically calculated energy gradients for BFGS optimization.

E. Classical numerical simulations

  • The simulations compare qubit and fermionic excitations and benchmark QEB-ADAPT-VQE on LiH, H6, and BeH2.LiH and BeH2 are commonly used for classical VQE-protocol benchmarks [40], while H6 models a strongly correlated ground state.Source: E. Classical numerical simulations · p. 9
  • A custom code supports arbitrary ansatz-element pools and growth strategies, with analytic excitation-based statevectors and Hamiltonian transformations using OpenFermion-psi4.The code second-quantizes the Hamiltonian before transforming it to quantum-gate-operator representation.Source: E. Classical numerical simulations · p. 9
  • The STO-3G basis [55, 56], without frozen orbitals, represents LiH, H6, and BeH2 with 12, 12, and 14 spin-orbitals and qubits, respectively.Source: E. Classical numerical simulations · p. 10
  • Variational parameters are optimized with BFGS using analytically calculated energy gradients; under high noise, gradient minimizers may struggle to find the global minimum [60].The passage notes that direct-search methods such as Nelder-Mead may perform better under high noise [63].Source: E. Classical numerical simulations · p. 10

F. Qubit versus fermionic excitations

Under a parameter-matched comparison, fermionic excitations converged faster than qubit excitations for LiH and BeH2 at large bond distances, with the largest gap for BeH2.

F. Qubit versus fermionic excitations

  • The protocols were compared with one variational parameter per qubit excitation versus one per individual fermionic excitation, isolating the effect of using qubit versus fermionic excitation pools.The QEB-ADAPT-VQE used n = 1 with step 5 not implemented; the fermionic-ADAPT-VQE appended individual excitations rather than spin-complement pairs.Source: F. Qubit versus fermionic excitations · p. 10
  • At 3 Å, fermionic-ADAPT-VQE used about 20% fewer ansatz elements on average than QEB-ADAPT-VQE to reach a given accuracy for BeH2; convergence also favored the fermionic protocol for LiH.The comparison used non-spin-complement fermionic excitation evolutions and QEB-ADAPT-VQE with n = 1 and step 5 not implemented.Source: Figure 4 · p. 11, F. Qubit versus fermionic excitations · p. 11
  • These results indicate that fermionic-excitation-based ansätze can approximate strongly correlated states better than qubit-excitation-based ansätze.The larger convergence-rate difference was observed for LiH and BeH2 at bond distances of 3 Å, with the clearest gap for BeH2.Source: F. Qubit versus fermionic excitations · p. 11

G. Energy dissociation curves

For LiH, H6, and BeH2, QEB-ADAPT-VQE reaches chemical accuracy across all investigated bond distances at the tighter energy-reduction thresholds, while its parameter efficiency depends on the molecule. The dissociation-curve comparison shows that strongly correlated cases challenge looser-threshold QEB-ADAPT-VQE and UCCSD-VQE.

G. Energy dissociation curves

  • At ϵ6 and ϵ8, QEB-ADAPT-VQE achieves chemical accuracy across all investigated bond distances for LiH, H6, and BeH2.Figure 5 uses n = 10 and step 5 in the STO-3G orbital basis set.Source: Figure 5 · p. 12, G. Energy dissociation curves · pp. 12–13
  • UCCSD-VQE achieves chemical accuracy across all LiH bond distances and near equilibrium for H6 and BeH2, but fails away from equilibrium for the latter two.The failures occur where the H6 and BeH2 ground states become more strongly correlated.Source: G. Energy dissociation curves · pp. 12–13, Figure 5 · p. 12
  • For LiH and BeH2, QEB-ADAPT-VQE at ϵ6 and ϵ8 is more accurate than UCCSD-VQE while using significantly fewer variational parameters; for H6 it uses more on average.H6 is more strongly correlated than LiH and BeH2, and even an optimally constructed ansatz would require more parameters than UCCSD to approximate its ground state accurately.Source: Figure 5 · p. 12, G. Energy dissociation curves · p. 13
  • For H6, QEB-ADAPT-VQE's parameter count changes abruptly near bond distances of 1 Å, 2 Å, and 2.75 Å, corresponding to molecular structure transformations and energy-level crossings.Different eigenstates of H become lowest in energy at these crossings.Source: Figure 5 · p. 12, G. Energy dissociation curves · p. 13

H. Energy convergence

The energy-convergence comparison shows QEB-ADAPT-VQE reducing CNOT cost relative to fermionic-ADAPT-VQE and outperforming qubit-ADAPT-VQE in selected metrics. Its performance depends on the metric and, against qubit-ADAPT-VQE, the target accuracy.

H. Energy convergence

  • The comparison uses unique single- and double-excitation pools and terminates convergence plots at 10^-12 Hartree.For qubit-ADAPT-VQE, the pool contains length-2 and length-4 XY Pauli strings with an odd number of Ys.Source: H. Energy convergence · pp. 13–14
  • At higher accuracies, QEB-ADAPT-VQE systematically uses fewer CNOTs than qubit-ADAPT-VQE and also beats fermionic-ADAPT-VQE in CNOT count.At low accuracies, qubit-ADAPT-VQE is more CNOT-efficient; qubit excitations allow local circuit optimizations that Pauli-string evolutions do not.Source: Figure 6 · p. 14, H. Energy convergence · p. 15
  • Compared with qubit-ADAPT-VQE, QEB-ADAPT-VQE systematically requires fewer iterations and variational parameters because qubit excitations combine 2 or 8 Pauli-string exponentials.The reported convergence plots cover LiH, H6, and BeH2.Source: Figure 6 · p. 14, H. Energy convergence · pp. 14–15
  • QEB-ADAPT-VQE and fermionic-ADAPT-VQE use similar iteration counts, but QEB-ADAPT-VQE can require up to twice as many variational parameters.QEB assigns one parameter per qubit excitation evolution, whereas fermionic-ADAPT-VQE assigns one per spin-complement pair.Source: Figure 6 · p. 14, H. Energy convergence · p. 15
  • An independent single-and-double fermionic pool converges more slowly but reaches higher final accuracy than the spin-complement-pair pool, owing to greater variational flexibility.Source: H. Energy convergence · p. 15

IV. DISCUSSION … Appendix D: Supplementary note 4

Qubit-excitation evolutions support molecular VQE ansätze with fixed-size circuits, and QEB-ADAPT-VQE combines them in an iterative, problem-tailored protocol. The discussion and supplementary notes describe its performance, tradeoffs, and computational costs alongside methods for calculating states and gradients.

IV. DISCUSSION

  • Qubit-excitation ansätze approximate molecular wavefunctions almost as accurately as fermionic-excitation ansätze, but fermionic ansätze are slightly more accurate per excitation for strongly correlated states.The discussion suggests parity-measuring Pauli-z strings play little role in variational flexibility; hardware-efficient ansätze and IQCC may be shallower but are described as unlikely to scale to large systems.Source: §IV · pp. 15–17
  • Compared with fermionic-ADAPT-VQE, QEB-ADAPT-VQE can require up to twice as many variational parameters but asymptotically fewer CNOTs; qubit excitations use fixed-size circuits.Fermionic excitation circuit size scales linearly in system size under Jordan-Wigner encoding and logarithmically under Bravyi-Kitaev encoding. Fermionic spin-complement pairs can enforce parity conservation and reduce variational parameters by up to a factor of 2.Source: §IV · p. 16
  • QEB-ADAPT-VQE systematically outperformed qubit-ADAPT-VQE in CNOT-efficiency for higher accuracies and required fewer variational parameters and ansatz-building iterations, though qubit-ADAPT-VQE was more efficient at low accuracies.Local circuit optimizations enabled by qubit evolutions are identified as the primary reason for the higher-accuracy advantage; the authors caution that larger-molecule benefits are suggested, not established generally.Source: §IV · p. 16
  • Potential QEB-ADAPT-VQE extensions include non-symmetry-preserving pool terms, pruning low-contribution evolutions, and estimating low-lying excited-state energies.The expanded pool could improve convergence and resilience to symmetry-breaking errors; pruning could further optimize the ansatz.Source: §IV · p. 17

Appendix A: Supplementary note 1

  • Appendix A derives the single-parameter energy-gradient expression in the main text using the skew-Hermiticity of the excitation operator.The derivation identifies the operator's skew-Hermiticity as the property that makes the intermediate expression equivalent to the main-text equation.Source: Appendix A · p. 17

Appendix B: Supplementary note 2

  • Appendix B precomputes fixed operator matrices and their squares to accelerate trial-state calculation; for BeH2 qubit-ADAPT-VQE, this used around 2GB of RAM and achieved nearly 20-fold speedup over Qiskit's general statevector simulator.The approach trades memory for faster statevector calculations.Source: Appendix B · p. 18

Appendix C: Supplementary note 3

  • Appendix C reports that calculating the energy gradient costs about three times a trial-state calculation, but using it reduces VQE optimization iterations by at least an order of magnitude.The numerical simulations calculate gradient components by updating intermediate statevectors to minimize matrix multiplications.Source: Appendix C · p. 19

Appendix D: Supplementary note 4

  • Appendix D says QEB-ADAPT-VQE step 2 is fully parallelizable, while the protocol's complexity depends on ansatz size and desired accuracy, which may vary with correlation strength.Parallel devices can reduce step 2 time to the cost of evaluating a single Pauli-string expectation value; initializing new ansatz parameters from the prior iteration can require fewer VQE evaluations than the stated worst case.Source: Appendix D · p. 20

Appendix E: Supplementary note 5

Testing larger candidate pools generally reduced QEB-ADAPT-VQE CNOT counts, but gains varied by molecule and saturated by n = 10; the benefit comes with more measurements and is justified when gate fidelity is the NISQ bottleneck.

Appendix E: Supplementary note 5

  • The n parameter sets how many qubit excitation evolutions are considered at each greedy ansatz-growth step, but noncommutation means the lowest-energy choice need not yield an optimal ansatz.Larger n raises the chance of selecting the evolution that achieves the largest energy reduction at each iteration.Source: Appendix E · p. 20
  • Classical energy-convergence simulations tested LiH, H6, and BeH2 ground states in STO-3G at 3 Å bond distances, away from equilibrium where correlation effects are stronger.The results are presented in Supplementary figure 7.Source: Appendix E · pp. 20–21, Figure 7 · p. 21
  • Using n > 1 offers a clear CNOT-count advantage despite increased quantum-computer measurement overhead, justified when quantum gate fidelity is the NISQ bottleneck.The authors expect the CNOT-count reduction to grow for larger molecules because their ansatz element pools are larger.Source: Appendix E · pp. 21–22

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